Uniform resolvent estimates for a non-dissipative Helmholtz equation

TL;DR

This paper investigates high frequency behavior of a non-dissipative Helmholtz equation, establishing resolvent estimates, eigenvalue absence, and analyzing semiclassical measures under specific damping and trapping conditions.

Contribution

It extends resolvent estimates and the limiting absorption principle to non-dissipative Helmholtz equations with trapped trajectories and weak damping assumptions.

Findings

Eigenvalues are absent on the upper half-plane near certain energies.

Resolvent estimates similar to Robert-Tamura are generalized.

Semiclassical measures are characterized for concentrated source terms.

Abstract

We study the high frequency limit for a non-dissipative Helmholtz equation. We first prove the absence of eigenvalue on the upper half-plane and close to an energy which satisfies a weak damping assumption on trapped trajectories. Then we generalize to this setting the resolvent estimates of Robert-Tamura and prove the limiting absorption principle. We finally study the semiclassical measures of the solution when the source term concentrates on a bounded submanifold of R^n.